# Caspar Wessel

Caspar Wessel
Born8 June 1745
Died25 March 1818 (aged 72)
Nationality Danish
Norwegian
Alma materUniversity of Copenhagen
Known forComplex numbers
Complex plane
Vectors
Partner(s)Cathrine Elisabeth Brinck f. Müller
AwardsKnight of the Order of the Dannebrog
Scientific career
FieldsMathematics

Caspar Wessel (8 June 1745, Vestby – 25 March 1818, Copenhagen) was a DanishNorwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the complex plane and vectors.[1]

## Biography

Wesselstua in Vestby, where Caspar and his brothers grew up

Wessel was born in Jonsrud, Vestby, Akershus in Norway to pastor Jonas Wessel (1707 - 85) and Helene Marie Schumacher (1715 - 89). Helene came from Drammen and was of Griffenfeldts heritage. Growing up in Wesselstuen in Vestby he was one of thirteen children in a family. His brothers include Johan Herman Wessel and Ole Christopher Wessel. Wessel is the grandnephew of the nobleman Peter Tordenskjold, brother in law of Maren Juel and the great-great-great-grandfather to the famous radio engineer Jan Wessel.[2]

In 1763, having completed secondary school at Oslo Cathedral School, he went to Denmark for further studies. He attended the University of Copenhagen to study law, but due to financial pressures, could do so for only a year. To survive, he became an assistant land surveyor to his brother and they worked on the Royal Danish Academy of Sciences and Letters' topographical survey of Denmark. This was not enough, however, and he took on extra work as a cartographer. He worked as a surveyor for the rest of his life, stopping only for a sabbatical year in 1778 to finish his law degree.[3] By 1798 Wessel had risen to the supervisory role of Royal Inspector of Surveying.[4]

It was the mathematical aspect of surveying that led him to exploring the geometrical significance of complex numbers. His fundamental paper, Om directionens analytiske betegning, was presented in 1797 to the Royal Danish Academy of Sciences and Letters.[5] Since it was in Danish and published in a journal rarely read outside of Denmark, it went unnoticed for nearly a century.[4] The same results were independently rediscovered by Jean-Robert Argand in 1806 and Carl Friedrich Gauss in 1831.[3]

"It is now Norway's cause to do what should be done, that this strange Man's Memory may be brought forth by Oblivion and his Name find its proper place before the History of Mathematics."

It was first when a Danish student wrote a doctoral dissertation on "Danish mathematics in the 18th century" that the universal significance of Wessel was brought up. While the student didn't realize the contribution by himself, the supervisors did. The dissertation created a great uproar and resulted in a front-page article in Dagbladet and Sophus Lie taking action to raise awareness of Wessel. In fact, Lie wrote in the article that "It is now Norway's cause to do what should be done, that this strange Man's Memory may be brought forth by Oblivion and his Name find its proper place before the History of Mathematics."[6]

One of the more prominent ideas presented in "On the Analytical Representation of Direction" was that of vectors. Even though this was not Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added together, when one joins them together so that one begins where the other one ends, and next one draws a straight line from the first to the last point of the joined lines, and takes this to be their sum". This is the same idea as used today in summing vectors.[7]

Memorial plaque of Wessel in Oldenburg, Germany

Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognised. His paper was re-issued in French translation in 1897,[8] and in English in 1999 as On the analytic representation of direction (eds. Branner & Lützen).[9]

From 1764 to 1805, Wessel participated in the survey of Denmark, Schleswig, Holstein and the Duchy of Oldenburg. His map of Schleswig-Holstein was requested by Napoleon Bonaparte.[10]

### Personal relations

Wessel had a modest and quiet personality, he was described by his brother Johan Herman as "He draws maps and reads the law. He's as diligent as I am lazy".[11] No portraits of Wessel seem to have survived.[7]

In 1780 he married Cathrine Elisabeth Brinck f. Müller in Copenhagen.[12]

On 1 July 1815, Wessel was made a knight of the Order of the Dannebrog for his contributions to surveying; the cross of the Order of the Dannebrog was presented to him at the Royal Palace of Amalienborg[where?] on 4 August 1815.[7]

## Sources

A Source Book in Mathematics, by David Eugene Smith, has "essential" passages of "Om directionens analytiske betegning" translated into English (see page 55).

## References

1. ^ "Caspar Wessel - Biography". Maths History. Retrieved 2021-05-11.
2. ^ Johansen, Nils Voje (2020-02-25), "Caspar Wessel", Norsk biografisk leksikon (in Norwegian Bokmål), retrieved 2021-04-27
3. ^ a b
4. ^ a b Nahin, Paul J. (1998). An Imaginary Tale: The Story of ${\displaystyle {\sqrt {-1}}}$. Princeton University Press. pp. 48–49. ISBN 978-0-691-14600-3.
5. ^ Wessel, Caspar (1799). "Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning" [On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons]. Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter (in Danish). Copenhagen: Royal Danish Academy of Sciences and Letters. 5: 469–518.
6. ^ Oslo, Universitetet i; Røsjø, Bjarne (2018-11-30). "Den glemte norske helten av matematikk: Caspar Wessel". forskning.no (in Norwegian). Retrieved 2021-04-27.
7. ^ a b c
8. ^ Wessel, Caspar (1799). Essai sur la représentation analytique de la direction [Essay on the Analytic Representation of Direction] (in French). Translated by Zeuthen, H. G. Copenhagen: Royal Danish Academy of Sciences and Letters (published 1897). BNF 31640182t.
9. ^ Wessel, Caspar (1797). Branner, Bodil; Lützen, Jesper (eds.). On the analytical representation of direction: an attempt applied chiefly to solving plane and spherical polygons, 1797. Translated by Damhus, Flemming. Copenhagen: C.A. Reitzels (published 1997). ISBN 8778761581. OCLC 43346556.
10. ^ "Caspar Wessel - Biography". Maths History. Retrieved 2021-05-11.
11. ^ Oslo, Universitetet i; Røsjø, Bjarne (2018-11-30). "Den glemte norske helten av matematikk: Caspar Wessel". forskning.no (in Norwegian). Retrieved 2021-04-27.
12. ^ Johansen, Nils Voje (2020-02-25), "Caspar Wessel", Norsk biografisk leksikon (in Norwegian Bokmål), retrieved 2021-04-27